Theorem natoppfb 50460

Description: A natural transformation is natural between opposite functors, and vice versa. (Contributed by Zhi Wang, 18-Nov-2025.)

Hypotheses

Ref	Expression
natoppf.o	⊢ 𝑂 = (oppCat‘𝐶)
natoppf.p	⊢ 𝑃 = (oppCat‘𝐷)
natoppf.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
natoppf.m	⊢ 𝑀 = (𝑂 Nat 𝑃)
natoppfb.k	⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
natoppfb.l	⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
natoppfb.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
natoppfb.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
natoppfb	⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))

Proof of Theorem natoppfb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		natoppf.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
2		natoppf.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐷)
3		natoppf.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		natoppf.m	. . . 4 ⊢ 𝑀 = (𝑂 Nat 𝑃)
5		natoppfb.k	. . . . 5 ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
6	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝐾 = ( oppFunc ‘𝐹))
7		natoppfb.l	. . . . 5 ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
8	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝐿 = ( oppFunc ‘𝐺))
9		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝑥 ∈ (𝐹𝑁𝐺))
10	1, 2, 3, 4, 6, 8, 9	natoppf2 50459	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝑥 ∈ (𝐿𝑀𝐾))
11		eqid 2760	. . . . 5 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂)
12		eqid 2760	. . . . 5 ⊢ (oppCat‘𝑃) = (oppCat‘𝑃)
13		eqid 2760	. . . . 5 ⊢ ((oppCat‘𝑂) Nat (oppCat‘𝑃)) = ((oppCat‘𝑂) Nat (oppCat‘𝑃))
14	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐿 = ( oppFunc ‘𝐺))
15	14	fveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐿) = ( oppFunc ‘( oppFunc ‘𝐺)))
16	4	natrcl 18135	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐿𝑀𝐾) → (𝐿 ∈ (𝑂 Func 𝑃) ∧ 𝐾 ∈ (𝑂 Func 𝑃)))
17	16	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐿 ∈ (𝑂 Func 𝑃) ∧ 𝐾 ∈ (𝑂 Func 𝑃)))
18	17	simpld 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐿 ∈ (𝑂 Func 𝑃))
19	14, 18	eqeltrrd 2861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐺) ∈ (𝑂 Func 𝑃))
20		relfunc 18044	. . . . . . 7 ⊢ Rel (𝑂 Func 𝑃)
21		eqid 2760	. . . . . . 7 ⊢ ( oppFunc ‘𝐺) = ( oppFunc ‘𝐺)
22	19, 20, 21	2oppf 50355	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘( oppFunc ‘𝐺)) = 𝐺)
23	15, 22	eqtr2d 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐺 = ( oppFunc ‘𝐿))
24	5	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐾 = ( oppFunc ‘𝐹))
25	24	fveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐾) = ( oppFunc ‘( oppFunc ‘𝐹)))
26	17	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐾 ∈ (𝑂 Func 𝑃))
27	24, 26	eqeltrrd 2861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))
28		eqid 2760	. . . . . . 7 ⊢ ( oppFunc ‘𝐹) = ( oppFunc ‘𝐹)
29	27, 20, 28	2oppf 50355	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘( oppFunc ‘𝐹)) = 𝐹)
30	25, 29	eqtr2d 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐹 = ( oppFunc ‘𝐾))
31		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐿𝑀𝐾))
32	11, 12, 4, 13, 23, 30, 31	natoppf2 50459	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐹((oppCat‘𝑂) Nat (oppCat‘𝑃))𝐺))
33	1	2oppchomf 17905	. . . . . . . 8 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))
34	33	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)))
35	1	2oppccomf 17906	. . . . . . . 8 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))
36	35	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (compf‘𝐶) = (compf‘(oppCat‘𝑂)))
37	2	2oppchomf 17905	. . . . . . . 8 ⊢ (Homf ‘𝐷) = (Homf ‘(oppCat‘𝑃))
38	37	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (Homf ‘𝐷) = (Homf ‘(oppCat‘𝑃)))
39	2	2oppccomf 17906	. . . . . . . 8 ⊢ (compf‘𝐷) = (compf‘(oppCat‘𝑃))
40	39	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (compf‘𝐷) = (compf‘(oppCat‘𝑃)))
41		natoppfb.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
42	41	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐶 ∈ 𝑉)
43		natoppfb.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
44	43	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐷 ∈ 𝑊)
45	1, 2, 42, 44, 27	funcoppc5 50368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐹 ∈ (𝐶 Func 𝐷))
46	45	func1st2nd 50297	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
47	46	funcrcl2 50300	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐶 ∈ Cat)
48	1	oppccat 17903	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
49	11	oppccat 17903	. . . . . . . 8 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat)
50	47, 48, 49	3syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (oppCat‘𝑂) ∈ Cat)
51	46	funcrcl3 50301	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐷 ∈ Cat)
52	2	oppccat 17903	. . . . . . . 8 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
53	12	oppccat 17903	. . . . . . . 8 ⊢ (𝑃 ∈ Cat → (oppCat‘𝑃) ∈ Cat)
54	51, 52, 53	3syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (oppCat‘𝑃) ∈ Cat)
55	34, 36, 38, 40, 47, 50, 51, 54	natpropd 18161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐶 Nat 𝐷) = ((oppCat‘𝑂) Nat (oppCat‘𝑃)))
56	3, 55	eqtrid 2807	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑁 = ((oppCat‘𝑂) Nat (oppCat‘𝑃)))
57	56	oveqd 7439	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐹𝑁𝐺) = (𝐹((oppCat‘𝑂) Nat (oppCat‘𝑃))𝐺))
58	32, 57	eleqtrrd 2863	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐹𝑁𝐺))
59	10, 58	impbida 813	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐹𝑁𝐺) ↔ 𝑥 ∈ (𝐿𝑀𝐾)))
60	59	eqrdv 2758	1 ⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ‘cfv 6541  (class class class)co 7422  1^st c1st 8001  2^nd c2nd 8002  Catccat 17845  Hom_f chomf 17847  comp_fccomf 17848  oppCatcoppc 17892   Func cfunc 18036   Nat cnat 18126   oppFunc coppf 50345

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-tpos 8248  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-er 8722  df-map 8854  df-ixp 8931  df-en 8979  df-dom 8980  df-sdom 8981  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-nn 12341  df-2 12410  df-3 12411  df-4 12412  df-5 12413  df-6 12414  df-7 12415  df-8 12416  df-9 12417  df-n0 12612  df-z 12699  df-dec 12820  df-sets 17349  df-slot 17367  df-ndx 17379  df-base 17395  df-hom 17459  df-cco 17460  df-cat 17849  df-cid 17850  df-homf 17851  df-comf 17852  df-oppc 17893  df-func 18040  df-nat 18128  df-oppf 50346

This theorem is used by:  fucoppclem  50649  lmddu  50957